A bicycle odometer (which measures distance travelepjqk07a8+k prd) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use8jp+arq kp07k it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim hv+.eb0 00),nju vsrc lh(p9sfj9 ml xsave radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the+cp)xen fsss vb. 09h9l muj00(rj,lv point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the f :4ib0zl.50u b gdmkyangular velocity $\omega$ Explain.

Can a small force ever exert a greaterqz22afe6 )np j torque than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-u 7tt des1p;; /qxsn6rpp with your hands behind your head than when your arms are stretched out in front of you? A diagram may het rqe/p17t s;d 6xnps;lp you to answer this.

A 21-speed bicycle has seven sprockets at the reargb1o,.vyzbv0hl45)c bs a,0 d 1zbb wg wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rea bb,s1y1lz.bo, )5b0vw vzhbc40a g dgr sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs whbac ea qc4+tv,q7l8;hmgpo4fz 9g*. ith flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain h c*4bfe7 9amhlv ,;q4pzagc+gq. o8 twhy this distribution of mass is advantageous.

Why do tightrope walkers (Fig.ga7lj x,8y+hgkjbjdkb9n); i -q0loal2i8m7 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the q w;3ct39 w-eh2hy wmmrtv qaq4yt412net torque on a- 4 ttqcqe3mr2;vy1w3th94 mw2 ahyqw system is zero, is the net force zero?

Two inclines have the samets wyws - t 6fak+wp,66/wg2xn height but make different angles with the horizontal. The same steel ball is rolled down each incli/2 tpxt ayf- 6s6ww+,6nww gksne. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously startf+pkwqowf2+w kl ;gzz 9zm6*( rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom zkk;(z o9 p+wmf+qwgz* fl62wof the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylindb9,5kjnh t1z*cd xa z-er have the same radius and the same mass. They start from rest at the top hkz jtb*n-az cdx1,59of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angularlz7 yu j;2lq(g+q-ag s momentum are conserved. Yet most moving or rotating objects eventually slow dg yq u;a-s7gqjz+ll(2own and stop. Explain.

If there were a great migration of people toward the Earth’sznytoa--)mah5qj / r (d7m6, a yca,du equator, how would this affect the lnaa,5 a m7 h,-my6- ur/yot q)z(dadjcength of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation wu.0id6i -zi bw w(ede3hen she leaves the boardii (u3 6.0-didwbee wz?

The moment of inertia of a jdwv 6a5wr;v20q 3oyjrotating solid disk about an axis through its center of j2jr 03yaw q v;dvow65mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating s6zi: x0ph vv3u 6 p-cian6kxy-tool holding a 2-kg mass in each outstretchedi-hc-:vk 0ppza 66xx 6yv ui3n hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and x jlx- kzxw;7yz98mf v2vbv;g 82p2iw the other is solid. Dww x;g x;vf9xzjbl28-v2v7mkp2 zy8iescribe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontalo h/ bv5.2yl4. nkxwbw axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is thhnb.kboy x5/w 2l .4wve angular speed increasing or decreasing?

Suppose you are standi5 g9ntdn8uo q2wn:3z lng on the edge of a large freely rotating turntable. What happens if you walk toward the cente2g n9twnl:dq5o8n u3zr?

A shortstop may leap into the air to catch sb- wvxl-1bqjf)7 gi0a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly yij sw f)0q-g1 b-lb7vxou will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation ofb 9ofo+bivjp 7f4e/y,y*xf w)vl6l w- angular momentum, discuss why a helicopter must have morbvw - 6v* ,+w/be ox)lfyjffyl9poi74e than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (alltp/rp ) q6+u) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coin67 p ijr7cd +p.hpwzay)+bkcq577 ym bcidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Mo7+z6)hw mky.5 p qycbjrppi 7d7cba+7on, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from90 rko2b0rx iv Earth. The beam diverges at an a2kvorx09i 0brngle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of c5lc) 5dulw uutgd8jq 0h4fy)u/-sa* 6500 rpm. When the motor is turned off during operation, the blawdtq5u-)) cl g*c5u uf0/hs8luadjy 4des slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makagq9:rjgq4t(zxk 6h +es 15.0 revolutions, what is its d +z:q4k6 r9q(x gjhgatiameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revoe*n s0zok+ :us o8*diolutions do the wheels make?08u*z n isoeos+d:*ko    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates atae j5 (2m7jbkc9o4yvl 2500 rpm. Calculate its angulara j4m79jy lvc2(o 5ebk velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Ffsv7te0dkz as - go)49ig. 8–38). (a) What is the linear speed of a childz 4t-sdfs0k ae97gvo) seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of:uygnom34+) s y)s gyl-km4ot the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speek3o mjq4qo5c 8d of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifugenwxbu*o-i/yi k gm3 cy,+7 (ie rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 10 w-u/( kyygxbiicon+* ,ei 3m70,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel loe i.+3yhsl(accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Deo se ll3.hiy+(termine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiuxv :4jf-m) 6 cwtl8a::o :o7nfy:yrcjaf ync8s $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, ash3., s6laq:zx1.t u6hn mxv mstronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, theysval xu h.t3,61sz:6qm . xhnm accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformlq kn2/zc 8)vdty from rest to 15,000 rpm in 220 s. Through how many revolutionn kc/)t8zdq2vs did it turn in this time?    $rev$

  An automobile engine slows down fxq6 mu2q6 8*5n2xkaarj1zo 8sp3 qutirom 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed 7ph3d2 psbccj*7-9 l2grvjrc jets in a whirling “human centrifuge,” which takes 1.0 min 2jr7jhd l7*c2g3 bcvs9c-r pp to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates unifoq4xlt bcz qgfh)76 60jrmly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in thi6lc6f xh70) tgq4jzqbs time?    m

  A cooling fan is turned off when it is running ajaqf,-wt y +m 8-h1ahz-w w4,ah5bby nt 850rev/min It turns 1500 revolutions before it comes to a stop. tzy4h aaf 5baq-wbn,-1+mw 8wh,y -hj
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 654zxlda.gc w+ k*- 2nij revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of gljkxc i*. -za n2w+d40.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed 3 yimhnt2j3a7 uniformly from 95km/h to 45km/h The tires have a diamih7 nat2m3y j3eter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weightk yt8w-ktkd*-mzu cia 8vk4,3+aqc9u on each pedal when climbing a hmdc zu,tiq a48ukkyk* 9+3wk ctv-a8- ill. The pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74n:.mshk me( x-wof4f5 cm wide. What is the magnitude of the torque .f swxe4-f5nmo h: (kmif the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Assumeer3d 4ep 38bll that a frictp38bl e4d rel3ion torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mak:0. pv+mt+rfwn3 :-xjeq fa dss m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude a+ ma n- +djwtxk:evqr0ff:3.p nd direction of the net torque on this system.

  The bolts on the cylinder head of a fjx9hj6pm 4hv,coz;7 n engine require tightening to a torque of 38j94 ,zxhf hj7opvc6 ;m $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sph y 38vczdw(nj;ere of radius 0.648 m when the axis of rotation isw;83cny ( vjzd through its center.    $kg \cdot m^2$

  Calculate the moment of aqx wrlcfqy)w 0wy:*65c su15inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass oyxwr f1q:5cq) w*lawuc 0y65sf 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is ro6 esa2xuxmu8f7 d9p9 /tayoy+91q4 i0ig hvgy tated in a horizontal circle of radius 1st7pyx uga 9+94x d2hv8oii9 0f a6ug e1yyqm/.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant anguljcf p :efvd qf,c6a1pxp4m- iwt: 14z/ar speed (Fig. 8–42). The friction force betwe6 /jc dc,mxvp4za -1iqw:e:t f1ffp 4pen her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point ob(x 0lhi dqb7*+mg ifs uab*)r1jects shown in Fig. 8–43 abouirbg1a0* qlh7 dbxf si(*)u+mt
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms wh (o8;sci v avoy;4trr-ose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindric4i y fkyaki7u) vc2jdh6+gd s2nh* (.lal satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.+yl*dfi c7s4 kn2 y. v)uhkd6iag j(h20 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with szy7abn*f; ws*ov0ak +my:ws g +1). 8cnvkisa radius of 8.50 cm and a mass of 0.580 kg. Calculc;f0gs:)yk+ na s ak8z+v *. y*swm svbo71winate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings+ krx9(.*+wq 9icgi kr4 lmcqc a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tan6.+sa y jxvhgcpue17+gentially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a 7+ cv6jsxpahu1g y +e.uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually brtmz-6xb t 1f (w7z xb ie4+g0np/aebm(ought uniformly to rest by a fricti (6in7t embxzw ( xaempb z01/bftg4-+onal torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ballxj;rd s9 )xj4o gu(vu. at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thf:g mhq:8sra .rown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly frs.haq:gmr8: fom rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a 2t w)ko:uhzlnsm 4q4 w6m.99pls jda2long thin rod, as shown in Fig. 8–46a :n m2qdm t. p24 lwkuzh)94l6wss9oj.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine cons m2svj;o(2 m;x2e mypiists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accm-c82*b*9 dofhzlx hhn2doz6 elerates the hammer from rest within four full turns (revolutions) and releases it atbdod9h xml2z h zco*6h 82f-n* a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertqj g1u-wo 09hxia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a t5:xy e7zxloo skte 90/orque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping downqtkk 09tz pqta:3u;r1:af. d b a lanf;btzu0k:tp k.qr3a 1 qa9t d:e at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect cx)c; . gcsps ,;dij9xto the Sun as the sum of two terms,x,i9 c j;)s p.cxgdcs;
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radiu,i8 dpe8zxtf, 1-enb5uj0x i cs of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolutionc8tzpi,e-edx u,nbf 8xi510 j per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from res5tls out9 7ilddm,+/nt and rolls without slippi7,um+l/9 odt l5sdnting down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its t zb+2p2n mfj d:xe(d5oip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of 2oj2pdxbn e+d5 zf:(mthe pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kgje rpdra0b4 p a+q/y.* ball rotating on the end of a thin string in a ci0 bqp.*da+ p 4rr/jeyarcle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical gribs5 b(fn6 si2 y- 95czc /jpbninwd(u+nding wheel of rasb+56pj-un5s(cb2i9 fdn/ zb(yi c wndius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is pf3ec5 h dk 4ui(n/y.cpl2wm; rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 82 klcpp/.w(ciu n e5mf;hy4d3–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can redxglgzy0y,y/1l1uu y ;uce her moment of inertia by a factor of about 3.5 when changing from th0yyu yxglg/;1l1yzu ,e straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase cp )qe;ve1x5w her spin rotation rate from an initial rate of 1.0 rev exec)qvp5 ;1w every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vert;2k. zgdv;ncsical axis through its center at a frequency of 1.5rev/s The wheel ca;;s.nzdv g2 ckn be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 35a vtosw( nl6- 3xb1bm.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum ov 4bbf(a5v- ikef-- krf the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an kp dsx47-cq/videntical disk rotating at angul/7k q vcdxps-4ar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a/ir1 e,rud/lt t 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the cente 8r prul5yyo0.ip4a+k r of a rotating merry-go-round platform of radius ua+pily8k 54op.rr 0 y3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular vehp-v bomhfs() v hme92ju;x4:locity of 02 usfp(hmb : ) 9-hmhvvjx4o;e.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventuallsh-p8 o(sq.ppz5k ha/ y collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular mqs(.p8-hs 5z p/kp ahoomentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excefi03teh ohp8q l.dlw 784wae,ss of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass +*nk,gwuaoj n-rqzvg074z.tc yk 3.*tyd zv 6 $ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initiallyucxp-by/y rn7 :)l3sw at rest, that can rotate freely without friction. The moment of inertia of the person plus the platfow7: b-3 /pulnryxcy) srm is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-round crrd*-z3r 6+( myiafwpia4j4 turntable that is mounted on frictionless bearings and has a momen3r4pari (+ 4i f6jz-cmadwyr *t of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope roll u ,b3lbs8mo*s*)vt x,5elqbss on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rl,eq)s sv *5t x,b *os8mlbb3uolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side amxl+ 9v2v : d1 gkf9n2zd,, koqzc zuwmg7xo4;lways faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter cao1+m ow 9gzd7zqnm,2xud,2klk9;fxvz4 g : vc se, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate ns- elv7t6:b: g iqhmo7(r7lwli .0l uof 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cyl6+ ucj +e6hse p 8lad*rw.l.a ice865 s)cunhdinder of radius 0.20 m acquires a rotationae). c8 cua p+ar6wd*s65 j.+hen6 cud8hsielll rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of to e3 ;dhm qn,4w)vt+bhwo solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (conc h,t 4hqwde +3;vm)nobentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed l(/s hqh, v9abof the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times k 5xyh:fv3u2*w-lvs nas great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gr2lvf-x :hwy3s u *5nkvavitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to uy,spyuta7z)*woh+ ( *a4*lty zs+zp ese energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km withoutuo 4p),( y+ayhspe*ts+a y z7*wtlzz* needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustra4+eng f4 dm;vi9h ws+ctes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizovy)i xgummr+ o (:j/ guy 7xy62 l1+r,zoaxhf.ntal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely sa )tuv nh6,k1bb ;im8(i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear accele8 ,m;bbt1uhs) ian6 kvration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It ix wiux9m: ig,-1n 53yqg+glgxj6zvy*s standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against j :5xyv +i36ggx*l g nzyimx1u-q9wg,which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn w :ez8b(bcpbo:4 o;mka ith a radius The forces acting on the cyzb:bbcpe4koa(;:o 8m clist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kgtom,efq46udx,f .1i )mp6nknjo /z*k rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal ci.6*xo) k z pjqnm/f mt61fdon4ki,,uercle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms ay+ yf.th ,c6ybs thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.+h6 bcy,y.tf y   

  You are designing a clutch aso xqkeye)z* :,sembly which consists of two cylindrical plates, of massez : eoqy)*xk, $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough xis a(5cul1kab)d, g9track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leal)a(xgk u5ia ,9sdcb1 ving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not a/cb+re mksstl +pk wqt .q6,(:ssume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduce)duib5k*m)s9v qwq1y s its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What wasmqdb*w uv9)q yk5s)1i the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s